- #1
skrat
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Homework Statement
Between two metal plates a metal strip is inserted as shown on the figure.
a) Calculate the electric potential anywhere inside the capacitor.
b) Simplifly ##U(x,y)## for ##x>>a##.
c) Calculate the electric field in the middle of the capactior.
Homework Equations
The Attempt at a Solution
a) Ok, since we have no other charges in the capacitor, than ##\nabla ^2 U=0##. In kartezian coordinates this means that ##U(x,y)=\sum _m (A_me^{-k_mx}+B_me^{k_mx})(Csin(k_my)+Dcos(k_my))##.
Using boundary conditions
##U(0,y)=U_0## and
##U(x,0)=0## and
##U(x,a)=0## we find out that
##U(x,y)=\sum _{n=1} ^{\infty}\frac{2U_0}{n\pi }(1-(-1)^n)e^{-\frac{n\pi}{a}x}\sin(\frac{n\pi }{a}y)##
b) The result is that out of the sum only ##n=1## survives, therefore ##U(x,y)=\frac{4U_0}{\pi }e^{-\frac{\pi}{a}x}\sin(\frac{\pi }{a}y)##.
I am having some troubles understanding this. Could anybody please try to explain how do I get this result?
c)
##U(x,a/2)=\sum _{n=1} ^{\infty}\frac{2U_0}{n\pi }(1-(-1)^n)e^{-\frac{n\pi}{a}x}\sin(\frac{n\pi }{2})##
##\vec E=-\nabla U(x,a/2)=\sum _{n=1} ^{\infty}\frac{2U_0}{a}(1-(-1)^n)\sin(\frac{n\pi }{2})e^{-\frac{n\pi}{a}x}##
I think I should do something more with that last equation but I really don't know what or how.. :/
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